Uniform periodic counterexamples to Carleson's convergence problem with polynomial symbols

Abstract

In Carleson's convergence problem for dispersive equations $i{\partial }_{t}u+P(D)u=0$ in the periodic setting ${\mathbb{T}}^d$, we prove that the Sobolev exponent $d/(2(d+1))$ is necessary for any non-singular polynomial symbol $P$, including the natural powers of the Laplacian $\hat{I}{”}^k$. This is in contrast with the results known in the Euclidean case, in which for symbols $P(\hat{I}¾)=|\hat{I}¾{|}^a$ with $a>1$ the exponent $d/(2(d+1))$ is sufficient, but we do not know if it is necessary.

v3: Manuscript rewritten with major modifications. Fractal result added